7

A

)

0 ¢

6¢ ‚

¢¢‚ 7 7

¢ u` YY

!

and , then the (column-) permuted linear system is

0 ‚ 00 ‚

34 @8 0 34 @89

9 34 @8 0

9

7 7 ¦

A 6 5 A ¢ 6 ‚ 66 6 ‚‚ 6 ‚‚ )0 ‚ 6

7

A

¦

5 5

7 ¦

¢¢ ‚ 6¢ ‚ ¢

7 7 ¦

¢

Note that only the unknowns have been permuted, not the equations, and in particular, the

right-hand side has not changed.

In the above example, only the columns of have been permuted. Such one-sided

permutations are not as common as two-sided permutations in sparse matrix techniques.

In reality, this is often related to the fact that the diagonal elements in linear systems play

a distinct and important role. For instance, diagonal elements are typically large in PDE

applications and it may be desirable to preserve this important property in the permuted

matrix. In order to do so, it is typical to apply the same permutation to both the columns

and the rows of . Such operations are called symmetric permutations, and if denoted by

¡" ¡

, then the result of such symmetric permutations satis¬es the relation

¡ G ¡ ¡ ¢

"¡

The interpretation of the symmetric permutation is quite simple. The resulting matrix cor-

responds to renaming, or relabeling, or reordering the unknowns and then reordering the

equations in the same manner.

¥ T ©

§¦

¥ ©

For the previous example, if the rows are permuted with the same permu-

tation as the columns, the linear system obtained is

‚‚ 0 0 ‚‚

34 34 @8 7

9 @8 0

9 34 @8 0

9

7

0 ¦

6‚ 7

)

0 A 6 5

¦

A ¢ 7‚ 6 67‚ 67‚

5 5 A6

6

7 ¦

¢¢ ‚ 6¢ ‚ ¢

7 7 ¦

¢

Observe that the diagonal elements are now diagonal elements from the original matrix,

placed in a different order on the main diagonal.

d nd¡ £ ¨¡ d¥ $ ¥ £ ”¤§¤

u © n ¥ u§¥ '¡

¤ §¥ ¡ ©

¨ £

¤©

¡ ¥

¥¤ ¤ ¦

B¡¥ ¥

A¡ !PG ' 1 ¥§

¨

#$ I¨ ' G' #% H

%

C© ¨

§( ' %

From the point of view of graph theory, another important interpretation of a symmetric

permutation is that it is equivalent to relabeling the vertices of the graph without altering

a`rB

the edges. Indeed, let be an edge in the adjacency graph of the original matrix

… ¡ ¡ ‚ p… ‚ aq`B rB wB B

C

"

and let be the permuted matrix. Then and a result is an edge

¦ ¦

C

in the adjacency graph of the permuted matrix , if and only if is an edge

¦

C

EC

C

in the graph of the original matrix . Thus, the graph of the permuted matrix has not

changed; rather, the labeling of the vertices has. In contrast, nonsymmetric permutations

do not preserve the graph. In fact, they can transform an indirected graph into a directed

one. Symmetric permutations may have a tremendous impact on the structure of the matrix

even though the general graph of the adjacency matrix is identical.

¥ © T © ©¨¦

§¥

Consider the matrix illustrated in Figure 3.4 together with its adjacency

graph. Such matrices are sometimes called “arrow” matrices because of their shape, but it

would probably be more accurate to term them “star” matrices because of the structure of

their graphs.

15)1)z p¥

If the equations are reordered using the permutation , the matrix and graph

shown in Figure 3.5 are obtained. Although the difference between the two graphs may

seem slight, the matrices have a completely different structure, which may have a signif-

icant impact on the algorithms. As an example, if Gaussian elimination is used on the

reordered matrix, no ¬ll-in will occur, i.e., the L and U parts of the LU factorization will